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Math 176 Calculus – Sec. 7.2: Trigonometric Integrals
I.
Evaluating
∫ sin
m
x cos n x dx
A. Strategy for Evaluating
∫ sin
m
x cos n x dx
(
)
1. If the power of cosine is odd n = 2k + 1 , save one cosine factor and use
cos x = 1 − sin x to express the remaining factors in terms of sine:
2
∫ sin
2
m
x cos 2 k +1 x dx = ∫ sin m x cos 2 k x cos x dx
(
)
x (1 − sin x )
= ∫ sin m x cos 2 x cos x dx
k
= ∫ sin m
k
2
cos x dx
Then substitute u= sinx.
2. If the power of sine is odd (m = 2k + 1), save one sine factor and use
sin 2 x = 1 − cos 2 x to express the remaining factors in terms of cosine:
∫ sin
2 k +1
x cos n x dx = ∫ sin 2 k x cos 2 k x sin x dx
∫ (sin x ) cos x sin xdx
= ∫ (1 − cos x ) cos x sin xdx
=
2
k
2
n
k
n
Then substitute u= cosx.
3. If the powers of both sine and cosine are odd, either 1 or 2 can be used.
4. If the powers of both sine and cosine are even, use the half-angle identities
1
[1 − cos(2 x )] and cos 2 x = 1 [1 + cos(2 x )]
2
2
1
5. It is sometimes helpful to use the identity sin x cos x = sin 2 x
2
sin 2 x =
B. Examples -- Evaluate the following
1.
∫ sin θ cos
4
θ dθ
2.
3.
∫ sin
∫
3
d
sin 2 x cos3 x
x
dx
4. Find the area bounded by the curve of
y = sin 4 (3x ) and the x-axis from x = 0 to
x=3.
II.
Evaluating
∫ tan
m
x sec n x dx
A. Strategy for Evaluating
∫ tan
m
x sec n x dx
1. If the power of secant is even
(n = 2k ) , save a factor of sec x and use
2
sec2 x = 1 + tan2 x to express the remaining factors in terms of tangent:
∫ tan
m
x sec2k x dx = ∫ tan m x sec2k −2 x sec2 x dx
( ) sec x dx
x (1 + tan x ) sec x dx
= ∫ tan m x sec 2 x
k −1
= ∫ tan m
2
Then substitute u= tanx.
2
k −1
2
2. If the power of tangent is odd
(m = 2k + 1) , save a factor of secxtanx and use
tan2 x = sec 2 x − 1 to express the remaining factors in terms of secant:
∫ tan
2k +1
x secn x dx = ∫ tan2k x secn−1 x sec x tan x dx
( )
= ∫ (sec x − 1) sec
k
= ∫ tan2 x secn−1 x sec x tan x dx
2
Then substitute u= secx.
B. Examples -- Evaluate the following
1.
∫ tan
6
y dy
6
2.
∫ tan
0
sec3
d
k
n−1
x sec x tan x dx
3. Find the volume of the solid generated by revolving the region bounded by the curve of
( )
( )
y = sec2 x 2 tan 2 x 2 , the x-axis and the line x =
III.
Evaluating
2
about the y-axis.
∫ tan x dx , ∫ cot x dx , ∫ sec x dx , ∫ csc x dx
A. Strategy for Evaluating
1. When evaluating
∫ tan x dx , ∫ cot x dx , ∫ sec x dx , ∫ csc x dx
∫ tan x dx
or
∫ cot x dx
rewrite the integrand in terms of sine and
cosine and then use u-substitution.
2. When evaluating
∫ sec x dx multiply the integrand by the form of one,
and then use u-substitution.
3. When evaluating
∫ csc x dx multiply the integrand by the form of one,
and then use u-substitution.
sec x + tan x
,
sec x + tan x
csc x + cot x
,
csc x + cot x
B. Examples
2
1.
∫
cot d
4
2.
∫ sec x dx
C. Integral Formulas for tan(x), cot(x), sec(x), csc(x)
1.
2.
3.
4.
∫ tan u du =
∫ cot u du =
∫ secu du =
∫ csc u du =
-lncos u + C = lnsec u + C
lnsin u + C = -ln cscu + C
lnsec u + tan u + C
-ln cscu + cot u + C